Convergence Rate Of Algorithm Research Articles

Influence of sampling on the convergence rates of greedy algorithms for parameter-dependent random variables

Influence of sampling on the convergence rates of greedy algorithms for parameter-dependent random variables

Joint sparse optimization: lower-order regularization method and application in cell fate conversion

Multiple measurement signals are commonly collected in practical applications, and joint sparse optimization adopts the synchronous effect within multiple measurement signals to improve model analysis and sparse recovery capability. In this paper, we investigate the joint sparse optimization problem via ℓp,q regularization ( 0⩽q⩽1⩽p ) in three aspects: theory, algorithm and application. In the theoretical aspect, we introduce a weak notion of joint restricted Frobenius norm condition associated with the ℓp,q regularization, and apply it to establish an oracle property and a recovery bound for the ℓp,q regularization of joint sparse optimization problem. In the algorithmic aspect, we apply the well-known proximal gradient algorithm to solve the ℓp,q regularization problems, provide analytical formulas for proximal subproblems of certain specific ℓp,q regularizations, and establish the global convergence and linear convergence rate of the proximal gradient algorithm under some mild conditions. More importantly, we propose two types of proximal gradient algorithms with the truncation technique and the continuation technique, respectively, and establish their convergence to the ground true joint sparse solution within a tolerance relevant to the noise level and the recovery bound under the assumption of restricted isometry property. In the aspect of application, we develop a novel method, based on joint sparse optimization with lower-order regularization and proximal gradient algorithm, to infer the master transcription factors for cell fate conversion, which is a powerful tool in developmental biology and regenerative medicine. Numerical results indicate that the novel method facilitates fast identification of master transcription factors, give raise to the possibility of higher successful conversion rate and in the hope of reducing biological experimental cost.

Decentralized intelligent multi-party competitive aggregation framework for electricity prosumers

Electricity management systems are experiencing significant challenges due to the increased penetration of distributed energy resources. Electricity flows in distribution networks are transforming from unidirectional to bi-directional form. Consumers are transitioning to prosumers with different characteristics, where they take more active roles in electricity generation and consumption. Aggregators are vital financial intermediary agents in the power system transitions, as they could aggregate energy profiles of prosumers. The market competition between aggregators and interactions between prosumers and aggregators are complex and dynamic, which requires a holistic framework to model the market competition. This paper proposes an intelligent aggregation framework with edge computing, enabling decentralized competition for multiple aggregators and prosumers, which can be solved with a graph-based consensus algorithm. This study mathematically proves the proposed algorithm's convergence guarantee and convergence rate. In addition, the proposed framework is applied to an open-source dataset to demonstrate its applicability. Lastly, a benchmark analysis is conducted to show that the proposed algorithm has better communication complexity than the benchmark algorithms.

A New Look and Convergence Rate of Federated Multitask Learning With Laplacian Regularization.

Non-independent and identically distributed (non-IID) data distribution among clients is considered as the key factor that degrades the performance of federated learning (FL). Several approaches to handle non-IID data, such as personalized FL and federated multitask learning (FMTL), are of great interest to research communities. In this work, first, we formulate the FMTL problem using Laplacian regularization to explicitly leverage the relationships among the models of clients for multitask learning. Then, we introduce a new view of the FMTL problem, which, for the first time, shows that the formulated FMTL problem can be used for conventional FL and personalized FL. We also propose two algorithms FedU and decentrali- zed FedU ( dFedU ) to solve the formulated FMTL problem in communication-centralized and decentralized schemes, respectively. Theoretically, we prove that the convergence rates of both algorithms achieve linear speedup for strongly convex and sublinear speedup of order 1/2 for nonconvex objectives. Experimentally, we show that our algorithms outperform the conventional algorithm FedAvg, FedProx, SCAFFOLD, and AFL in FL settings, MOCHA in FMTL settings, as well as pFedMe and Per-FedAvg in personalized FL settings.

Hardware-algorithm co-design of Energy Efficient Federated Learning in Quantized Neural Network

The real-world implementation of federated learning (FL) for smart device-based applications necessitates energy-efficient convergent models due to resource-constrained devices. In the literature, deep learning (DL)–based models are frequently trained on FL to obtain high accuracy and quick convergence employing a 32-bit floating point precision level. However, such circumstances are not feasible for devices with limited resources. DL models demand a significant amount of processing power and energy, which has a noticeable negative impact on the environment. Hence, it is necessary to reduce the level of precision in DL models to enhance energy efficiency. In this paper, we propose a low-precision level green federated learning (GFL) model to maintain the balance of communication cycles, energy efficiency, and accuracy with an admissible convergence rate of the deep learning algorithms. Our proposed GFL framework achieves 98.04% server accuracy and 97.69% federated accuracy with a faster convergence rate and fewer communication rounds than state-of-the-art methods on the UCI human activity recognition dataset.

Convergence Rate of Algorithm for Solving Linear Equations by Quantum Annealing

Convergence Rate of Algorithm for Solving Linear Equations by Quantum Annealing

Integrating human learning and reinforcement learning: A novel approach to agent training

Off-policy reinforcement learning (RL) algorithms are known for improving sample efficiency by employing prior experiences in experience replay memory. However, most existing off-policy RL algorithms are bottlenecked by the slow convergence speed and demand of a large number of interaction samples. In contrast, on-policy RL algorithms converge fast and continuously generate new samples. However, their success heavily depends on the accuracy of the generated samples. To address these challenges, a novel RL framework called HI-FER (Human-learning Inspired Frequent Experience Replay) is proposed by mimicking the process of human learning. HI-FER employs a parallelized experience replay and repetitive training framework to expedite the convergence rate of off-policy algorithms, which imitates the function of the human brain’s parallel information processing and repetitive learning. Additionally, a periodic network reset strategy and dynamic memory updating are leveraged by imitating the forgetting mechanism of humans to prevent overfitting triggered by repetitive updating on limited experiences. Extensive comparison experiments and ablation studies are performed on benchmark environments to evaluate the proposed method. The empirical results demonstrate that HI-FER outperforms the baselines in terms of sample efficiency on state-based (14% improvements) and image-based (51% improvements) tasks from DMControl. Project website and code: https://github.com/Arya87/HI-FER.

Decentralized Sum-of-Nonconvex Optimization

We consider the optimization problem of minimizing the sum-of-nonconvex function, i.e., a convex function that is the average of nonconvex components. The existing stochastic algorithms for such a problem only focus on a single machine and the centralized scenario. In this paper, we study the sum-of-nonconvex optimization in the decentralized setting. We present a new theoretical analysis of the PMGT-SVRG algorithm for this problem and prove the linear convergence of their approach. However, the convergence rate of the PMGT-SVRG algorithm has a linear dependency on the condition number, which is undesirable for the ill-conditioned problem. To remedy this issue, we propose an accelerated stochastic decentralized first-order algorithm by incorporating the techniques of acceleration, gradient tracking, and multi-consensus mixing into the SVRG algorithm. The convergence rate of the proposed method has a square-root dependency on the condition number. The numerical experiments validate the theoretical guarantee of our proposed algorithms on both synthetic and real-world datasets.

Learning Korobov Functions by Correntropy and Convolutional Neural Networks.

Combining information-theoretic learning with deep learning has gained significant attention in recent years, as it offers a promising approach to tackle the challenges posed by big data. However, the theoretical understanding of convolutional structures, which are vital to many structured deep learning models, remains incomplete. To partially bridge this gap, this letter aims to develop generalization analysis for deep convolutional neural network (CNN) algorithms using learning theory. Specifically, we focus on investigating robust regression using correntropy-induced loss functions derived from information-theoretic learning. Our analysis demonstrates an explicit convergence rate for deep CNN-based robust regression algorithms when the target function resides in the Korobov space. This study sheds light on the theoretical underpinnings of CNNs and provides a framework for understanding their performance and limitations.

Large-eddy simulation-based reconstruction of turbulence in a neutral boundary layer using spectral-tensor regularization

We propose an efficient method to reconstruct the turbulent flow field in a neutrally stratified atmospheric boundary layer using large-eddy simulation (LES) and a series of lidar measurements. The reconstruction is formulated as a strong four-dimensional variational data assimilation problem, which involves optimizing two competing terms that contribute in the objective functional. The first term is a likelihood term, while the second contains the initial background distribution of turbulent velocity fluctuations and works as a regularization term. However, computing and storing the full background covariance tensor in turbulent flows is time consuming and resource intensive. In the current work, we investigate the possibility of replacing the complex background tensor by simple analytical approximations based on spectral tensors such as the Hunt–Graham–Wilson (HGW) model (Boundary-Layer Meteorol., vol. 85, 1997, pp. 35–52) or the Mann model (J. Fluid Mech., vol. 273, 1994, pp. 141–168). Afterwards, the problem is solved using a quasi-Newton algorithm and preconditioned to enhance the convergence rate. We test the method using virtual lidar measurements collected on a fine reference LES. Results show a super-linear convergence rate of the optimization algorithm to a local minimum and very good agreement between virtual lidar measurements and reconstruction in the scanning region. Furthermore, we demonstrate that incorporating the Saffman energy spectrum ( $E(k) \sim k^2$ where E is the energy spectrum and k is the magnitude of the wavenumber vector) at low wavenumbers into the Mann spectral tensor yields a longer streamwise correlation length, resulting in reduced reconstruction error when compared with the Batchelor spectrum ( $E(k) \sim k^4$ ). Finally, we observe that using the HGW model or Mann model with a Saffman spectrum yields similar results.

Entropy-based convergence rates of greedy algorithms

We present convergence estimates of two types of greedy algorithms in terms of the entropy numbers of underlying compact sets. In the first part, we measure the error of a standard greedy reduced basis method for parametric PDEs by the entropy numbers of the solution manifold in Banach spaces. This contrasts with the classical analysis based on the Kolmogorov [Formula: see text]-widths and enables us to obtain direct comparisons between the algorithm error and the entropy numbers, where the multiplicative constants are explicit and simple. The entropy-based convergence estimate is sharp and improves upon the classical width-based analysis of reduced basis methods for elliptic model problems. In the second part, we derive a novel and simple convergence analysis of the classical orthogonal greedy algorithm for nonlinear dictionary approximation using the entropy numbers of the symmetric convex hull of the dictionary. This also improves upon existing results by giving a direct comparison between the algorithm error and the entropy numbers.

On constrained Kaczmarz algorithm with momentum for image reconstruction

SummaryThe Kaczmarz algorithm is widely used in image reconstruction. In this paper, we propose an enhanced version of the Kaczmarz algorithm by integrating Nesterov momentum into the Kaczmarz algorithm, denoted as the Kaczmarz algorithm with momentum. Our innovation offers a novel approach to accelerate the convergence rate of the Kaczmarz algorithm significantly. Based on greedy rules for selecting work row, we prove that the Kaczmarz method with momentum can achieve global linear convergence rates. Furthermore, we extend a constrained version of the Kaczmarz algorithm with Nesterov momentum for improved image reconstruction and prove its convergence. Numerical experiments show that the proposed algorithm outperforms the original Kaczmarz algorithm in terms of efficiency for image reconstruction.

A modified white shark optimizer for optimal power flow considering uncertainty of renewable energy sources

This paper presents a novel approach to solve the optimal power flow (OPF) problem by utilizing a modified white shark optimization (MWSO) algorithm. The MWSO algorithm incorporates the Gaussian barebones (GB) and quasi-oppositional-based learning (QOBL) strategies to improve the convergence rate and accuracy of the original WSO algorithm. To address the uncertainty associated with renewable energy sources, the IEEE 30 bus system, which consists of 30 buses, 6 thermal generators, and 41 branches, is modified by replacing three thermal generators with two wind generators and one solar PV generator. And the IEEE 57-bus system, which consists of 57 buses, 7 thermal generators, and 80 branches, is also modified by the same concept. The variability of wind and solar generation is described using the Weibull and lognormal distributions, and its impact on the OPF problem is considered by incorporating reserve and penalty costs for overestimation and underestimation of power output. The paper also takes into account the unpredictability of power consumption (load demand) by analyzing its influence using standard probability density functions (PDF). Furthermore, practical conditions related to the thermal generators, such as ramp rate limits are examined. The MWSO algorithm is evaluated and analyzed using 23 standard benchmark functions, and a comparative study is conducted against six well-known techniques using various statistical parameters. The results and statistical analysis demonstrate the superiority and effectiveness of the MWSO algorithm compared to the original WSO algorithm for addressing the OPF problem in the presence of generation and demand uncertainties.

A Particle Swarm and Smell Agent-Based Hybrid Algorithm for Enhanced Optimization

The particle swarm optimization (PSO) algorithm is widely used for optimization purposes across various domains, such as in precision agriculture, vehicular ad hoc networks, path planning, and for the assessment of mathematical test functions towards benchmarking different optimization algorithms. However, because of the inherent limitations in the velocity update mechanism of the algorithm, PSO often converges to suboptimal solutions. Thus, this paper aims to enhance the convergence rate and accuracy of the PSO algorithm by introducing a modified variant, which is based on a hybrid of the PSO and the smell agent optimization (SAO), termed the PSO-SAO algorithm. Our specific objective involves the incorporation of the trailing mode of the SAO algorithm into the PSO framework, with the goal of effectively regulating the velocity updates of the original PSO, thus improving its overall performance. By using the trailing mode, agents are continuously introduced to track molecules with higher concentrations, thus guiding the PSO’s particles towards optimal fitness locations. We evaluated the performance of the PSO-SAO, PSO, and SAO algorithms using a set of 37 benchmark functions categorized into unimodal and non-separable (UN), multimodal and non-separable (MS), and unimodal and separable (US) classes. The PSO-SAO achieved better convergence towards global solutions, performing better than the original PSO in 76% of the assessed functions. Specifically, it achieved a faster convergence rate and achieved a maximum fitness value of −2.02180678324 when tested on the Adjiman test function at a hopping frequency of 9. Consequently, these results underscore the potential of PSO-SAO for solving engineering problems effectively, such as in vehicle routing, network design, and energy system optimization. These findings serve as an initial stride towards the formulation of a robust hyperparameter tuning strategy applicable to supervised machine learning and deep learning models, particularly in the domains of natural language processing and path-loss modeling.

A recent proximal gradient algorithm for convex minimization problem using double inertial extrapolations

<abstract><p>In this study, we suggest a new class of forward-backward (FB) algorithms designed to solve convex minimization problems. Our method incorporates a linesearch technique, eliminating the need to choose Lipschitz assumptions explicitly. Additionally, we apply double inertial extrapolations to enhance the algorithm's convergence rate. We establish a weak convergence theorem under some mild conditions. Furthermore, we perform numerical tests, and apply the algorithm to image restoration and data classification as a practical application. The experimental results show our approach's superior performance and effectiveness, surpassing some existing methods in the literature.</p></abstract>

Optimal Period Input Design in FIR System Identification With Binary-Valued Observations and Event-Triggered Communication

In the identification of networked systems, the sender uses the event-triggered mechanism to reduce the times of sending data to save communication resources, but the receiver needs as much data as possible to improve the estimation accuracy. This paper tries to mediate the contradiction between the two by means of input design. First, based on the communication rate and the convergence rate of the identification algorithm, the problem of optimal input design is became an optimization problem with constraints. Then, the algorithm for solving this optimization problem is given. Next, the two-time-scale implementation method of optimal input is proposed, where the updated time of the parameter estimation constitutes the identified time scale, and the input is updated based on the estimated value of the system parameter and lasts for a period of time that constitutes the duration time scale of the input. Finally, the validity of the algorithm is verified by numerical simulation.

Improved convergence rates for some kernel random forest algorithms

<abstract><p>Random forests are notable learning algorithms introduced by Breiman in 2001. They are widely used for classification and regression tasks and their mathematical properties are under ongoing research. We consider a specific class of random forest algorithms related to kernel methods, the so-called Kernel Random Forests (KeRF). In particular, we investigate thoroughly two explicit algorithms, designed independently of the data set, the centered KeRF and the uniform KeRF. In the present article, we provide an improvement in the rate of convergence for both algorithms and we explore the related reproducing kernel Hilbert space defined by the explicit kernel of the centered random forest.</p></abstract>

Continuous Control With Swarm Intelligence Based Value Function Approximation

Value function approximation, such as Q-learning, is widely used in the discrete control rather than the continuous one because the optimal action in the discrete setting is more easily selected. Optimizing the action is a non-convex optimization problem with respect to the complex value function. Some notable studies simplify the non-convex optimization problem by assuming the value function as quadratic in the actions or by discretizing the action space. However, the performance of the output policy will decline if these studies’ premises do not hold. In order to address the problem, we propose a framework that combines swarm intelligence algorithms with value-based Reinforcement Learning, where the swarm intelligence algorithms are employed to search for the optimal action with respect to the state and the value function. To ensure the correctness of this framework, we conditionally claim the convergence rate of swarm intelligence algorithms with high probability. We then implement it by searching the batch optimal actions to various states on the GPU platform for the batch training. Furthermore, we employ the population-based atomic actions for the compatibility with the existing related work about solving discrete control problems. Four classical control models and four robot simulation environments are utilized in the comparisons. According to empirical results, our framework outputs a policy comparable with that of the policy-based algorithms by 10% timesteps in the continuous control. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper is motivated by the exploration-exploitation dilemma of Reinforcement Learning to solve continuous control tasks. To balance the exploration and exploitation, the stochastic exploration and the prioritized exploration are roughly two feasible ways, where the prioritized one is a better choice due to the higher data efficiency than the stochastic one, e.g. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon$</tex-math> </inline-formula> -greedy. Normally, the prioritized exploration works well in the value-based Reinforcement Learning algorithms rather than the policy-based ones; meanwhile, the policy-based algorithms are more suitable to continuous control tasks than the value-based ones. To tackle this conflict, we especially design a particle swarm optimization to maximize the Q-value of action in Q-learning. Our design can be hybridized by various swarm intelligence and value-based Reinforcement Learning algorithms. Also, it can be embedded in most intelligent control systems easily. The aim of this study is to solve the continuous control tasks by value-based algorithms as the first step of applying the prioritized exploration. The simulative results verify the effectiveness and efficiency of our design.

Generalized nesterov’s algorithm for constrained minimization problems on closed convex sets

In this paper, we propose a generalized Nesterov algorithm for the constrained optimization problems on a closed convex set. We prove the convergence as well as the convergence rate of the proposed algorithm. First, we present a new algorithm based on the generalization of Nesterov’s algorithm. Then, we prove the convergence as well as the convergence rate of the new algorithm. With a specific choice of parameters, the new algorithm becomes Nesterov’s algorithm. Therefore, the convergence as well as the convergence rate of Nesterov’s algorithm are also followed. We illustrate the effectiveness of the new algorithm as well as compare it with Nesterov’s algorithm and the gradient descent algorithm through a specific example.

Improved artificial bee colony algorithm based on two-dimensional queue structure for complex optimization problems

In the face of increasingly complex optimization problems, the Artificial Bee Colony algorithm still encounters some unresolved issues. For instance, the randomness involved in generating the initial population provides diversity to optimization solutions but also leads to convergence among individuals due to the current pseudo-randomness in this process. To address this challenge, this study introduces an improved ABC algorithm based on a two-dimensional queue structure, referred to as KLABC. The KLABC algorithm introduces several key advantages over traditional ABC algorithms. By employing the K-Means algorithm, it enhances the assessment of initial population individuals, ensuring a higher-quality population that aligns with specified requirements. The integration of a two-dimensional queue-based search strategy, along with the application of the first-in, first-out (FIFO) mechanism, not only preserves population diversity but also minimizes the risk of discarding potential optimal solutions during iterative searches. Augmenting this framework, the concepts of the global best solution (GBest) and queue best solution (QBest) are incorporated to fortify the exploration formula, expediting the algorithm's convergence rate. Experimental results on clustering optimization problems across seven UCI datasets validate the effectiveness and robustness of the proposed algorithm. Experimental results indicate that the KLABC algorithm achieves faster convergence compared to the traditional ABC algorithm, with an average performance improvement of 55.63%. Additionally, when comparing the runtime of different algorithms in handling large-scale data, the KLABC algorithm effectively maintains its performance even in scenarios with larger data sizes.